Speaking in Glasgow, where his party finished third in three constituencies, Mr Sheridan said the SSP was on course for more than 70,000 votes across Scotland.
Mr Sheridan, who before polling day predicted his party would secure 100,000 votes, nevertheless described the result as "historic" and said: "Scotland now has five political parties."
" What we didn't foresee was the huge drop in turnout which is a badge of shame for the four big business parties "
He blamed the other four parties for the low turnout among voters, claiming they had failed to convince the electorate.
And the SSP leader criticised the media for what he claimed was a "disgraceful" lack of coverage during the campaign.
Mr Sheridan accused the BBC of "politically censoring" the SSP.
He said that the party had not managed to gain the 100,000 votes it had hoped for before the election campaign began.
"We have not realised the 100,000 target, which we knew was ambitious, but what we didn't foresee was the huge drop in turnout which is a badge of shame for the four big business parties," Mr Sheridan said.
"They have had wall-to-wall, cover-to-cover TV and newspaper coverage, and yet they have failed to inspire the electorate of Glasgow and Scotland.
"Despite the political censorship of our campaign by the broadcast media and the daily broadsheets, we have secured a magnificent 70,000 votes for genuine, radical socialism and we feel we have now arrived on the political scene in Scotland."
Mr Sheridan went on to predict his party would break through the 100,000 barrier at the next Scottish Parliament elections in 2003 and return eight MSPs to Holyrood.
" In the next two years, we will build a party that will deliver not one MSP, but eight, one for every region in Scotland "
He said: "We suffer from the fact that this is a Westminster election and we're a Scottish party, so some people will think they should vote for a Westminster-wide party. I accept that.
"I think we will surpass 100,000 votes in Scotland next time. We have gone from 40,000 votes in 1999, the highest we had ever achieved, to 70,000 votes in two years.
"In the next two years, we will build a party that will deliver not one MSP, but eight, one for every region in Scotland."
SSP candidates finished third in Glasgow Springburn - where Carolyn Leckie stood against the Commons Speaker, Michael Martin, Glasgow Pollok and Glasgow